C321 Nielsen's Generalized Polylogarithm

Function subprograms CGPLG and WGPLG calculate the complex-valued generalized polylogarithm function

$S$_n,m(x) = {(-1)^n+m-1(n-1)! m!} ∫₀¹t^-1ln^n-1t  ln^m(1-xt) dt (*)

for real arguments x and integer n and m satisfying $1\le n\le 4,\; 1\le m\le 4,\; n+m\le 5$ ; i.e., one of the functions $S$_1,1 , $S$_1,2 , $S$_2,1 , $S$_1,3 , $S$_2,2 , $S$_3,1 , $S$_1,4 , $S$_2,3 , $S$_3,2 , $S$_4,1 . If $x\le 1$ , $S$_n,m(x) is real, and the imaginary part is set equal to zero.

The double-precision version WGPLG is available only on computers which support a COMPLEX*16 Fortran data type.

Structure:

FUNCTION subprograms
User Entry Names: CGPLG, WGPLG
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

In any arithmetic expression, CGPLG(N,M,X) or WGPLG(N,M,X) has the value $S$_N,M(X) ,

where CGPLG is of type COMPLEX, WGPLG is of type COMPLEX*16, X is of type REAL for CGPLG and of type DOUBLE PRECISION for WGPLG, and where N and M are of type INTEGER.

Method:

The method is described in Ref. 1. Note that the imaginary part of the function defined as $S$_n,m(x)

in Ref. 1 has the opposite sign to the imaginary part of the function defined by (*). See Ref. 2.

Accuracy:

CGPLG (except on CDC and Cray computers) has full single-precision accuracy. For most values of the argument X, WGPLG (and CGPLG on CDC and Cray computers) has an accuracy of approximately two significant digits less than the machine precision. The loss of accuracy is greater when X is very close to 1.

Error handling:

Error C321.1: or $N,M\; >\; 4$ or $N+M\; >\; 5$ . The function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

References:

1. K.S. Kölbig, J.A. Mignaco and E. Remiddi, On Nielsen's generalized polylogarithms and their numerical calculation, BIT 10 (1970) 38--71.
2. K.S. Kölbig, Nielsen's generalized polylogarithms, SIAM J. Math. Anal. 17 (1986) 1232--1258.

C322